We present new extensions of the two-step, triangular-pattern phase-shifting method for different numbers of phase-shifting steps to increase measurement accuracy and to analyze the influence of the number of phase-shifting steps and pitch of the projected triangular intensity-profile pattern on the measurement accuracy. Phase-shifting algorithms to generate the intensity ratio, essential for surface reconstruction, were developed for each measurement method. Experiments determined that higher measurement accuracy can be obtained with a greater number of phase-shifting steps and a lower value of pitch, as long as the pitch is appropriately selected to be divisible by the number of phase-shifting steps and not below an optimal value, where intensity-ratio unwrapping failure would occur.

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In phase-shifting-based fringe-projection surface-geometry measurement, phase unwrapping techniques produce a continuous phase distribution that contains the height information of the 3-D object surface. Mapping of the phase distribution to the height of the object has often involved complex derivations of the nonlinear relationship. In this paper, the phase-to-height mapping is formulated using both linear and nonlinear equations, the latter through a simple geometrical derivation. Furthermore, the measurement accuracies of the linear and nonlinear calibrations are compared using measurement simulations where noise is included at the calibration stage only, and where noise is introduced at both the calibration and measurement stages. Measurement accuracies for the linear and nonlinear calibration methods are also compared, based on real-system measurements. From the real-system measurements, the accuracy of the linear calibration was similar to the nonlinear calibration method at the lower range of depth. At the higher range of depth, however, the nonlinear calibration method had considerably higher accuracy. It seems that as the object approaches the projector and camera for the higher range of depth, the assumption of linearity based on small divergence of light from the projector becomes less valid.

We present an intensity-ratio error-compensation method to decrease the measurement error caused by projector gamma nonlinearity and image defocus in triangular-pattern phase-shifting profilometry. The intensity-ratio measurement error is first determined by simulating the measurement with the triangular-pattern phase-shifting method with ideal and real captured triangular-pattern images based on the ideal and real gamma nonlinearity functions. A lookup table that stores the intensity-ratio measurement error corresponding to the measured intensity ratio is constructed and used for intensity-ratio error compensation. Experiments demonstrated that the intensity-ratio error compensation method significantly reduced the measurement error in the triangular-pattern phase-shifting method by 28.5%.

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